Problem 2: 2D Stability of a SIRS Epidemic In epidemiology, the SIRS model (Susceptible-Infected-Recovered-Susceptible) describes the temporal dynamics of the outbreak of a disease that the majority of the population can recover from (without dying) and then become susceptible to again at some point in the future such as the flu. The model is described by the following nonlinear system of ODEs dS dt dl dt dR dt dS dldR dt dt dt I. Express the only non-trivial steady state of the system in terms of , , , and P. To clarify, two other steady states exist (S = P, 1-0, R = 0) and (S-0, 1-0, R-P). Find the only non-trivial steady state 2. Let 0.0005, -0.1, = 0.001, and P 500 . Express the linearized system of equations centered at the equilibrium in terms of the Jacobian Matrix. as as as S- S aS al OR Write out the secular determinant of the Jacobian detJ-11-0. Then, use the eigQ function in Matlab to find the eigenvalues and eigenvectors of the Jacobian Matrix Comment on the stability based on the eigenvalues Simulate the system over the time interval t = [0, 5000] with initial condition (S-499, I 3. 4. 1, R = 0) using the parameters defined above. Plot the time course and a 3D phase portrait using plot30) Problem 2: 2D Stability of a SIRS Epidemic In epidemiology, the SIRS model (Susceptible-Infected-Recovered-Susceptible) describes the temporal dynamics of the outbreak of a disease that the majority of the population can recover from (without dying) and then become susceptible to again at some point in the future such as the flu. The model is described by the following nonlinear system of ODEs dS dt dl dt dR dt dS dldR dt dt dt I. Express the only non-trivial steady state of the system in terms of , , , and P. To clarify, two other steady states exist (S = P, 1-0, R = 0) and (S-0, 1-0, R-P). Find the only non-trivial steady state 2. Let 0.0005, -0.1, = 0.001, and P 500 . Express the linearized system of equations centered at the equilibrium in terms of the Jacobian Matrix. as as as S- S aS al OR Write out the secular determinant of the Jacobian detJ-11-0. Then, use the eigQ function in Matlab to find the eigenvalues and eigenvectors of the Jacobian Matrix Comment on the stability based on the eigenvalues Simulate the system over the time interval t = [0, 5000] with initial condition (S-499, I 3. 4. 1, R = 0) using the parameters defined above. Plot the time course and a 3D phase portrait using plot30)